Good point - though it has me wondering this: the equi-spaced grid is a case having a straightforward proof (given that Cantor did the heavy lifting), but is it a necessary condition?
The proof relies on the fact that the rationals are lesser in cardinality than the reals, but for each "rational point" from your field of view, there are infinite trees in that line. So anything anything which doesn't force all those trees into a rational point is going to cover something larger than the rationals... and then you'll probably need an answer to the continuum hypotheses?